$A_{\infty }$-algebras associated with curves and rational functions on $\mathcal{M}_{g, g}$. I. (April 2014) - Fisette, Robert
Abstract We consider the natural $A_{\infty }$ -structure on the $\mathrm{Ext}$ -algebra $\mathrm{Ext}^*(G, G)$ associated with the coherent sheaf $G=\mathcal{O}_C\oplus \mathcal{O}_{p_1}\oplus \cdots \oplus \mathcal{O}_{p_n}$ on a smooth projective curve $C$ , where $p_1, \ldots, p_n\in C$ are distinct points. We study the homotopy class of the product $m_3$ . Assuming that $h^0(p_1+\cdots +p_n)=1$ , we prove that $m_3$ is homotopic to zero if and only if $C$ is hyperelliptic and the points $p_i$ are Weierstrass points. In the latter case we show that $m_4$ is not homotopic to zero, provided the genus of $C$ is greater than $1$ . In the case $n=g$ we prove that the $A_{\infty }$ -structure is determined uniquely (up to homotopy) by the products $m_i$ with $i\le 6$ . Also, in this case we study the rational map $\mathcal{M}_{g, g}\to \mathbb{A}^{g^2-2g}$ associated with the homotopy class of $m_3$ . We prove that for $g\ge 6$ it is birational onto its image, while for $g\le 5$ it is dominant. We also give an interpretation of this map in terms of tangents to $C$ in the canonical embedding and in the projective embedding given by the linear series $|2(p_1+\cdots +p_g)|$ .